Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T02:35:12.831Z Has data issue: false hasContentIssue false

Elliptic curves with good reduction away from 2

Published online by Cambridge University Press:  24 October 2008

R. G. E. Pinch
Affiliation:
Mathematics Department, University of Glasgow, Glasgow G12 8QW

Extract

In this paper we list the elliptic curves defined over Q √ − 1, Q√ −2 or Q√ − 3 which have good reduction away from 2. The possible invariants of such curves are given in Table 1, and their minimal equations in Tables 2, 3 and 4. These extend (and agree with) results of Ogg[4] and Stroeker [10], by a different method.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Birch, B. J. and Kuyk, W. (eds.). Modular Functions of One Variable. IV. Lecture Notes in Math. vol. 476 (Springer, 1975).CrossRefGoogle Scholar
[2] Hassb, H.. Number Theory, Grundlehren der math. Wissenschaften 229 (Springer, 1980).CrossRefGoogle Scholar
[3] Klein, F. and Fricke, R.. Vorlesungen uber die Theorie der elliptischen Modulfunktionen (Stuttgart, Teubner, 1966).Google Scholar
[4] Ogg, A.. Abclian curves of 2-power conductor. Proc. Cambridge Philos. Soc. 62 (1966), 143148.CrossRefGoogle Scholar
[5] Ogg, A.. Elliptic curves and wild ramification. Amer. J. Math. 89 (1967), 121.CrossRefGoogle Scholar
[6] Pinch, R. G. E.. Elliptic curves over number fields. D.Phil. Thesis, Oxford, 1982.Google Scholar
[7] Pinch, R. G. E.. Computing with algebraic integers, in preparation.Google Scholar
[8] Pohst, M.. On the computation of number fields of small discriminants. J. Number Theory 14 (1982), 99117.CrossRefGoogle Scholar
[9] Serre, J.-P. and Tate, J.. Good reduction of Abelian varieties. Ann. Math 88 (1968), 492527.CrossRefGoogle Scholar
[10] Stroeker, R.. Elliptic curves denned over imaginary quadratic number fields. Ph.D.Thesis, Amsterdam, 1975.Google Scholar
[11] Tate, J.. Algorithm for defining the type of a singular fibre in an elliptic pencil. In [1] 3352.CrossRefGoogle Scholar
[12] Weiss, E.. Algebraic Number Theory (Chelsea, 1963).Google Scholar